Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(11,15)(14,16)(17,18,21,24)(19,23,20,22)(26,30)(27,33)(28,35)(38,40);(1,3)(2,7)(4,5)(6,8)(17,18)(19,22)(20,23)(21,24)(27,33)(31,36)(34,39)(38,40);(2,4)(3,8)(9,10,13,12)(11,14,15,16)(17,19,21,20)(18,22,24,23)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,4,6,2)(3,5,8,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,23)(20,22)(21,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,5)(2,8)(3,4)(6,7)(9,12,13,10)(11,16,15,14)(17,20,21,19)(18,23,24,22)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=fh,c^2=hi,d^2=fgh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2667549);